A misclassification cost‐minimizing evolutionary–neural classification approach

Machine learning algorithms that incorporate misclassification costs have recently received considerable attention. In this paper, we use the principles of evolution to develop and test an evolutionary/genetic algorithm (GA)‐based neural approach that incorporates asymmetric Type I and Type II error costs. Using simulated, real‐world medical and financial data sets, we compare the results of the proposed approach with other statistical, mathematical, and machine learning approaches, which include statistical linear discriminant analysis, back‐propagation artificial neural network, integrated cost preference‐based linear mathematical programming‐based minimize squared deviations, linear integrated cost preference‐based GA, decision trees (C 5.0, and CART), and inexpensive classification with expensive tests algorithm. Our results indicate that the proposed approach incorporating asymmetric error costs results in equal or lower holdout sample misclassification cost when compared with the other statistical, mathematical, and machine learning misclassification cost‐minimizing approaches. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

Cost-based due-date assignment with the use of classical and neural-network approaches

Traditional methods of due-date assignment presented in the literature and used in practice generally assume cost-of-earliness and cost-of-tardiness functions that may bear little resemblance to true costs. For example, practitioners using ordinary least-squares (OLS) regression implicitly minimize a quadratic cost function symmetric about the due date, thereby assigning equal second-order costs to early completion and tardy behavior. In this article the consequences of such assumptions are pointed out, and a cost-based assignment scheme is suggested whereby the cost of early completion may differ in form and/or degree from the cost of tardiness. Two classical approaches (OLS regression and mathematical programming) as well as a neural-network methodology for solving this problem are developed and compared on three hypothetical shops using simulation techniques. It is found for the cases considered that: (a) implicitly ignoring cost-based assignments can be very costly; (b) simpler regression-based rules cited in the literature are very poor cost performers; (c) if the earliness and tardiness cost functions are both linear, linear programming and neural networks are the methodologies of choice; and (d) if the form of the earliness cost function differs from that of the tardiness cost function, neural networks are statistically superior performers. Finally, it is noted that neural networks can be used for a wide range of cost functions, whereas the other methodologies are significantly more restricted. © 1997 John Wiley & Sons, Inc.     

Lagrangian relaxation approach to the targeting problem

In this paper, we consider a new weapon–target allocation problem with the objective of minimizing the overall firing cost. The problem is formulated as a nonlinear integer programming model. We applied Lagrangian relaxation and a branch‐and‐bound method to the problem after transforming the nonlinear constraints into linear ones. An efficient primal heuristic is developed to find a feasible solution to the problem to facilitate the procedure. In the branch‐and‐bound method, three different branching rules are considered and the performances are evaluated. Computational results using randomly generated data are presented. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 640–653, 1999     

Statistical decision analysis of stochastic linear programming problems

This paper presents a statistical decision analysis of a one-stage linear programming problem with deterministic constraints and stochastic criterion function. Procedures for obtaining numerical results are given which are applicable to any problem having this general form. We begin by stating the statistical decision problems to be considered, and then discuss the expected value of perfect information and the expected value of sample information. In obtaining these quantities, use is made of the distribution of the optimal value of the linear programming problem with stochastic criterion function, and so we discuss Monte Carlo and numerical integration procedures for estimating the mean of this distribution. The case in which the random criterion vector has a multivariate Normal distribution is discussed separately, and more detailed methods are offered. We discuss dual problems, including some relationships of this work with other work in probabilistic linear programming. An example is given in Appendix A showing application of the methods to a sample problem. In Appendix B we consider the accuracy of a procedure for approximating the expected value of information.     

A squared-euclidean distance location-allocation problem

This article is concerned with the analysis of a squared-Euclidean distance location-allocation problem with balanced transportation constraints, where the costs are directly proportional to distances and the amount shipped. The problem is shown to be equivalent to maximizing a convex quadratic function subject to transportation constraints. A branch-and-bound algorithm is developed that utilizes a specialized, tight, linear programming representation to compute strong upper bounds via a Lagrangian relaxation scheme. These bounds are shown to substantially dominate several other upper bounds that are derived using standard techniques as problem size increases. The special structure of the transportation constraints is used to derive a partitioning scheme, and this structure is further exploited to devise suitable logical tests that tighten the bounds implied by the branching restrictions on the transportation flows. The transportation structure is also used to generate additional cut-set inequalities based on a cycle prevention method which preserves a forest graph for any partial solution. Results of the computational experiments, and a discussion on possible extensions, are also presented.     

More ado about economic order quantities (EOQ)

This paper is concerned with the determination of explicit expressions for economic order quantities and reorder levels, such that the cost of ordering and holding inventory is minimized for specific backorder constraints. Holding costs are applied either to inventory position or on-hand inventory, and the backorder constraint is considered in terms of the total number of backorders per year or the average number of backorders at any point in time. Through the substitution of a new probability density function in place of the normal p.d.f., explicit expressions are determined for the economic order quantities and the reorder points. The resulting economic order quantities are independent of all backorder constraints. It is also concluded that under certain conditions, the minimization of ordering costs and inventory holding costs (applied to inventory position), subject to a backorder constraint, is equivalent in terms of reorder levels to minimization of the safety level dollar investment subject to the same backorder constraint.     

A linear-programming-based method for determining whether or not n demand points are on a hemisphere

Whenever n demand points are located on a hemisphere, spherical location problems can be solved easily using geometrical methods or mathematical programming. A method based on a linear programming formulation with four constraints is presented to determine whether n demand points are on a hemisphere. The formulation is derived from a modified minimax spherical location problem whose Karush-Kuhn-Tucker conditions are the constraints of the linear program. © 1993 John Wiley & Sons, Inc.     

A descent approach to solving the complementary programming problem

In recent years, much attention has focused on mathematical programming problems with equilibrium constraints. In this article we consider the case where the constraints are complementarity constraints. Problems of this type arise, for instance, in the design of traffic networks. We develop here a descent algorithm for this problem that will converge to a local optimum in a finite number of iterations. The method involves solving a sequence of subproblems that are linear programs. Computational tests comparing our algorithm with the branch-and-bound algorithm in [7] bear out the efficacy of our method. When solving large problems, there is a definite advantage to coupling both methods. A local optimum incumbent provided by our algorithm can significantly reduce the computational effort required by the branch-and-bound algorithm.     

Models for multi-item continuous review inventory policies subject to constraints

Models are formulated for determining continuous review (Q, r) policies for a multiitem inventory subject to constraints. The objective function is the minimization of total time-weighted shortages. The constraints apply to inventory investment and reorder workload. The formulations are thus independent of the normal ordering, holding, and shortage costs. Two models are presented, each representing a convex programming problem. Lagrangian techniques are employed with the first, simplified model in which only the reorder points are optimized. In the second model both the reorder points and the reorder quantities are optimized utilizing penalty function methods. An example problem is solved for each model. The final section deals with the implementation of these models in very large inventory systems.     

An integer programming algorithm for portfolio selection with fixed charges

A mean-variance portfolio selection model with limited diversification is formulated in which transaction and management costs are incorporated as the sum of a linear cost and a fixed cost. The problem is a fixed charge integer programming problem solved by hypersurface search using dynamic programming. Fathoming is performed in the forward pass of dynamic programming so that values of the state variable which correspond to infeasible solutions are eliminated from the tables. This logic permits the solution of problems with 20–30 possible investments.     

A single-machine replacement model with learning

The replacement or upgrade of productive resources over time is an important decision for a manufacturing organization. The type of technology used in the productive resources determines how effectively the manufacturing operations can support the product and marketing strategy of the organization. Increasing operating costs (cost of maintenance, labor, and depreciation) over time force manufacturing organizations to periodically consider replacement or upgrade of their existing productive resources. We assume that there is a setup cost associated with the replacement of a machine, and that the setup cost is a nonincreasing function of the number of replacements made so far due to learning in setups. The operating cost of a newer machine is assumed to be lower than the operating cost of an older machine in any given period, except perhaps in the first period of operation of the new machine when the cost could be unusually high due to higher initial depreciation. A forward dynamic programming algorithm is developed which can be used to solve finite-horizon problems. We develop procedures to find decision and forecast horizons such that choices made during the decision horizon based only on information over the forecast horizon are also optimal for any longer horizon problem. Thus, we are able to obtain optimal results for what is effectively an infinite-horizon problem while only requiring data over a finite period of time. We present a numerical example to illustrate the decision/forecast horizon procedure, as well as a study of the effects of considering learning in making a series of machine replacement decisions. © 1993 John Wiley & Sons. Inc.     

Variations on a cutting plane method for solving concave minimization problems with linear constraints

A cutting plane method for solving concave minimization problems with linear constraints has been advanced by Tui. The principle behind this cutting plane has been applied to integer programming by Balas, Young, Glover, and others under the name of convexity cuts. This paper relates the question of finiteness of Tui's method to the so-called generalized lattice point problem of mathematical programming and gives a sufficient condition for terminating Tui's method. The paper then presents several branch-and-bound algorithms for solving concave minimization problems with linear constraints with the Tui cut as the basis for the algorithm. Finally, some computational experience is reported for the fixed-charge transportation problem.     

An explicit general solution in linear fractional programming

A complete analysis and explicit solution is presented for the problem of linear fractional programming with interval programming constraints whose matrix is of full row rank. The analysis proceeds by simple transformation to canonical form, exploitation of the Farkas-Minkowki lemma and the duality relationships which emerge from the Charnes-Cooper linear programming equivalent for general linear fractional programming. The formulations as well as the proofs and the transformations provided by our general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases. The augmentation developing the explicit solution is presented, for clarity, in an algorithmic format.     

The cost of production breaks

Disruptions in the production process can have a serious impact on production costs. Most of the previous literature which addresses the cost impact of production breaks attributes the observed increase in costs solely to “loss of learning.” We develop a mathematical model which implies that breaks in the unit learning curve can occur because of a production break, even under the assumption of no forgetting. In such cases, increases in unit cost can be caused by decreasing returns as the amount of time available to meet the delivery schedule decreases due to the production break.     

Rank‐Cluster‐and‐Prune: An algorithm for generating clusters in complex set partitioning problems

Clustering problems are often difficult to solve due to nonlinear cost functions and complicating constraints. Set partitioning formulations can help overcome these challenges, but at the cost of a very large number of variables. Therefore, techniques such as delayed column generation must be used to solve these large integer programs. The underlying pricing problem can suffer from the same challenges (non‐linear cost, complicating constraints) as the original problem, however, making a mathematical programming approach intractable. Motivated by a real‐world problem in printed circuit board (PCB) manufacturing, we develop a search‐based algorithm (Rank‐Cluster‐and‐Prune) as an alternative, present computational results for the PCB problem to demonstrate the tractability of our approach, and identify a broader class of clustering problems for which this approach can be used. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

线性规划优化分析的元模型方法及其比较

线性规划优化分析在经济管理等领域有着广泛的应用。当线性规划约束条件的右端向量在一定范围内变化时,目标函数的最优值是右端向量的一个复杂的分片线性函数,但通常难以给出分析表达式。应用多项式回归、径向基函数、Kriging法及多项式回归 Kriging法这四种元模型方法,能快速预测最优值函数。通过仿真实验,对这四种形式的元模型作较全面的比较分析。数值实验的结果表明,用次数较少的实验设计,后三种方法都具有较高的拟合精度;特别地,多项式回归 Kriging法不仅拟合精度高,而且还能用一个二阶多项式给出最优值函数的一个简明的近似描述。结果表明,元模型方法是研究线性规划优化分析问题的有效途径。     

Analysis of the turbulent regime of the progress curve when new learning additions have variable slopes

Learning curves have been used extensively for predictive purposes in the airframe and other industries. In many instances this has led to erroneous results because analysts failed to extend learning curve theory and develop adequate analytical techniques in the turbulent regime of the cost history characterizing these industries. It is this area where a series of design changes induces a series of perturbations whose turbulence intensity is a function of the frequency of occurrence and magnitudes of the design changes under consideration. In Ref. 1, a series of formulations amenable to machine programming was developed for the accurate determination of perturbed unit costs. This development was based on additions of new learning having a constant slope. In this discussion, the development of Ref. 1 will be generalized by developing formulas for the addition of new effort having variable slopes. Consideration will also be given to the expressions involving elementary unit cost expressions so that cumulative average and cumulative total values can readily be obtained from existing experience curve tables. Conversely, the problem of determining the magnitudes of design changes and the slopes of new effort from graphical data will also be considered.     

Comparisons of renewable warranties

Two types of warranties are commonly applied. There are replacement‐free warranty and pro‐rata warranty. The third warranty is the mixing of the above two. That is it has a replacement‐free stage followed by a pro‐rata stage. In addition to the above classification, warranties can also be a renewable or nonrenewable. No matter which warranty is used, it always incurs certain cost to the manufacturer. Moreover, often manufacturers want to improve the quality of products by burn‐in before delivering them to consumers, and this incurs an extra cost. This article considers the random costs for both burn‐in and renewable warranty, and derives the mean warranty cost. Then the mean costs corresponding to pure replacement‐free warranty, mixing renewable warranty, and pure pro‐rata warranty are compared. Furthermore, the mean costs under the same warranty policy associated with different lifetime distributions of products are also compared. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 91–106, 1999     

A note on duality in homogeneous fractional programming

For a linear fractional programming problem, Sharma and Swarup have constructed a dual problem, also a linear fractional program, in which the objective functions of both primal and dual problems are the same. Craven and Mond have extended this result to a nonlinear fractional programming problem with linear constraints, and a dual problem for which the objective function is the same as that of the primal. This theorem is now further extended from linear to differentiable convex constraints.